group theory

# Contents

## Definition

For a prime number $p$, the Prüfer $p$-group is defined uniquely up to isomorphism as the group where every element has exactly $p$ ${p}^{\mathrm{th}}$ roots. It is a divisible abelian group which can be described in several ways, for example:

• It is the discrete group that is Pontryagin dual to the compact topological group of p-adic integers.

• It is $ℤ\left[1/p\right]/ℤ$, the colimit of the sequence of inclusions

$0↪ℤ/\left(p\right)↪\dots ↪ℤ/\left({p}^{n}\right)↪ℤ/\left({p}^{n+1}\right)↪\dots$0 \hookrightarrow \mathbb{Z}/(p) \hookrightarrow \ldots \hookrightarrow \mathbb{Z}/(p^n) \hookrightarrow \mathbb{Z}/(p^{n+1}) \hookrightarrow \ldots

## Properties

The Prüfer $p$-groups are the only infinite groups whose subgroups are totally ordered by inclusion. They are often useful as counterexamples in algebra; for example, a Prüfer group is an Artinian but not a Noetherian $ℤ$-module.

Revised on January 31, 2012 20:30:50 by Urs Schreiber (82.169.65.155)